Paper 2011/127

Integer Arithmetic without Arithmetic Addition

Gideon Samid

Abstract

Revisiting long established conventions has proven very fertile in many a case. Let’s then revisit the premise that arithmetic must be constructed with the arithmetic addition as its foundation. Here we explore an arithmetic realm over integers without invoking the quintessential operation of addition. We propose an arithmetic constructed over a fundamental mapping of one set of integers into another. We start and focus here on mapping an arbitrary number of integers to a single integer, and further limit our investigation to a mapping procedure that views the input integers as a set of conflicting answers to a binary question, and attempt to figure out the single integer that best reflects the combined “wisdom” of the input answers. Thereby we construct the proposed arithmetic as ground tool for discriminant analysis. On the other end, the many-to-one mapping suggests this arithmetic as a fundamental hashing function, and the complexity of data loss suggests a new primitive for asymmetric cryptography. This arithmetic evolved from practical algorithms used by the author in his engineering practice, where the original name was BiPSA: Binary Polling Scenario Analysis. For continuity purposes we carry on the name. This article focuses on the skeleton arithmetic. Applications and substantiation will follow.

Note: Fixing some lines in the PDF file.